Last Time
• Visibility
– Area Subdivision (Warnock’s)
– BSP Trees
– Cells and Portals
3/25/04
© University of Wisconsin, CS559 Spring 2004
Today
• Lighting and Shading
• No lecture Tuesday March 30
3/25/04
© University of Wisconsin, CS559 Spring 2004
Where We Stand
• So far we know how to:
– Transform between spaces
– Draw polygons
– Decide what’s in front
• Next
– Deciding a pixel’s intensity and color
3/25/04
© University of Wisconsin, CS559 Spring 2004
Normal Vectors
• The intensity of a surface depends on its orientation with
respect to the light and the viewer
• The surface normal vector describes the orientation of the
surface at a point
– Mathematically: Vector that is perpendicular to the tangent plane of
the surface
• What’s the problem with this definition?
– Just “the normal vector” or “the normal”
– Will use n or N to denote
• Normals are either supplied by the user or automatically
computed
3/25/04
© University of Wisconsin, CS559 Spring 2004
Transforming Normal Vectors
• Normal vectors are directions
Normal vectors are perpedicul ar to tangent v ectors : n  (x  p)  0
There is a matrix form of this : n t (x  p)  0
Consider t he equation w ith a transform ed tangent : n t T 1T(x  p)  0
The right hand half is the transform ed point.
The new transpose normal must be equal to : n t T 1
The new normal must then be : (n t T 1 ) t  (T 1 ) t n
• To transform a normal, multiply it by the inverse transpose of the
transformation matrix
• Recall, rotation matrices are their own inverse transpose
• Don’t include the translation! Use (x,y,z,0) for homogeneous coordinates
3/25/04
© University of Wisconsin, CS559 Spring 2004
Local Shading Models
• Local shading models provide a way to determine the
intensity and color of a point on a surface
– The models are local because they don’t consider other objects
– We use them because they are fast and simple to compute
– They do not require knowledge of the entire scene, only the current
piece of surface. Why is this good?
• For the moment, assume:
– We are applying these computations at a particular point on a
surface
– We have a normal vector for that point
3/25/04
© University of Wisconsin, CS559 Spring 2004
Local Shading Models
• What they capture:
– Direct illumination from light sources
– Diffuse and Specular reflections
– (Very) Approximate effects of global lighting
• What they don’t do:
–
–
–
–
3/25/04
Shadows
Mirrors
Refraction
Lots of other stuff …
© University of Wisconsin, CS559 Spring 2004
“Standard” Lighting Model
• Consists of three terms linearly
combined:
– Diffuse component for the amount of
incoming light reflected equally in all
directions
– Specular component for the amount of
light reflected in a mirror-like fashion
– Ambient term to approximate light
arriving via other surfaces
3/25/04
© University of Wisconsin, CS559 Spring 2004
Diffuse Illumination
kd I i ( L  N )
• Incoming light, Ii, from direction L, is reflected equally in all directions
– No dependence on viewing direction
• Amount of light reflected depends on:
– Angle of surface with respect to light source
• Actually, determines how much light is collected by the surface, to then be
reflected
– Diffuse reflectance coefficient of the surface, kd
• Don’t want to illuminate back side. Use
3/25/04
kd I i max( L  N ,0)
© University of Wisconsin, CS559 Spring 2004
Diffuse Example
Where is the light?
Which point is brightest
(how is the normal at the
brightest point related to
the light)?
3/25/04
© University of Wisconsin, CS559 Spring 2004
Illustrating Shading Models
• Show the polar graph of the amount of light leaving for a given
incoming direction:
Diffuse?
• Show the intensity of each point on a surface for a given light position
or direction
Diffuse?
3/25/04
© University of Wisconsin, CS559 Spring 2004
Illustrating Shading Models
• Show the polar graph of the amount of light leaving for a given
incoming direction:
Diffuse:
• Show the intensity of each point on a surface for a given light position
or direction
Diffuse:
3/25/04
© University of Wisconsin, CS559 Spring 2004
Specular Reflection
(Phong Model)
L
V
R
k s I i (R  V )
p
• Incoming light is reflected primarily in the mirror direction,
R
– Perceived intensity depends on the relationship between the viewing
direction, V, and the mirror direction
– Bright spot is called a specularity
• Intensity controlled by:
– The specular reflectance coefficient, ks
– The Phong Exponent, p, controls the apparent size of the specularity
• Higher n, smaller highlight
3/25/04
© University of Wisconsin, CS559 Spring 2004
Specular Example
3/25/04
© University of Wisconsin, CS559 Spring 2004
Illustrating Shading Models
• Show the polar graph of the amount of light leaving for a given
incoming direction:
Specular?
• Show the intensity of each point on a surface for a given light position
or direction
Specular?
3/25/04
© University of Wisconsin, CS559 Spring 2004
Illustrating Shading Models
• Show the polar graph of the amount of light leaving for a given
incoming direction:
Specular:
Specular Lobe
• Show the intensity of each point on a surface for a given light position
or direction
Specular:
3/25/04
© University of Wisconsin, CS559 Spring 2004
Specular Reflection
Improvement
H  L  V  / L  V
L H N V
k s I i (H  N) p
• Compute based on normal vector and “halfway” vector, H
– Always positive when the light and eye are above the tangent plane
– Not quite the same result as the other formulation (need 2H)
3/25/04
© University of Wisconsin, CS559 Spring 2004
Putting It Together

I  k a I a  I i k d (L  N )  k s ( H  N ) p

• Global ambient intensity, Ia:
– Gross approximation to light bouncing around of all other surfaces
– Modulated by ambient reflectance ka
• Just sum all the terms
• If there are multiple lights, sum contributions from each light
• Several variations, and approximations …
3/25/04
© University of Wisconsin, CS559 Spring 2004
Color
I r  ka ,r I a ,r  Ii ,r kd ,r ( L  N )  ks,r ( H  N )
• Do everything for three colors, r, g and b
• Note that some terms (the expensive ones) are constant
• For reasons we will not go into, using only three colors is an
approximation, but few graphics practitioners realize it
– Aliasing in color space
– Better results use 9 color samples
3/25/04
© University of Wisconsin, CS559 Spring 2004
n

Approximations for Speed
• The viewer direction, V, and the light direction, L, depend
on the surface position being considered, x
• Distant light approximation:
– Assume L is constant for all x
– Good approximation if light is distant, such as sun
• Distant viewer approximation
– Assume V is constant for all x
– Rarely good, but only affects specularities
3/25/04
© University of Wisconsin, CS559 Spring 2004
Distant Light Approximation
• The viewer direction, V, and the light direction, L, depend
on the surface position being considered, x
• Distant light approximation:
– Assume L is constant for all x
– Good approximation if light is distant, such as sun
– Generally called a directional light source
• What aspects of surface appearance are affected by this
approximation?
– Diffuse?
– Specular?
3/25/04
© University of Wisconsin, CS559 Spring 2004
Local Viewer Approximation
• Specularities require the viewing direction:
– V(x) = ||c-x||
– Slightly expensive to compute
• Local viewer approximation uses a global V
– Independent of which point is being lit
– Use the view plane normal vector
– Error depends on the nature of the scene
• Is the diffuse component affected?
3/25/04
© University of Wisconsin, CS559 Spring 2004
Describing Surfaces
• The various parameters in the lighting equation describe the appearance
of a surface
• (kd,r,kd,g,kd,b): The diffuse color, which most closely maps to what you
would consider the “color” of a surface
– Also called diffuse reflectance coefficients
• (ks,r,ks,g,ks,b): The specular color, which controls the color of specularities
– The same as the diffuse color for metals, white for plastics
– Some systems do not let you specify this color separately
• (ka,r,ka,g,ka,b): The ambient color, which controls how the surface looks
when not directly lit
– Normally the same as the diffuse color
3/25/04
© University of Wisconsin, CS559 Spring 2004
OpenGL Model
•
•
•
•
•
•
•
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Allows emission, E: Light being emitted by surface
Allows separate light intensity for diffuse and specular
Ambient light can be associated with light sources
Allows spotlights that have intensity that depends on outgoing light
direction
Allows attenuation of light intensity with distance
Can specify coefficients in multiple ways
Too many variables and commands to present in class
The OpenGL programming guide goes through it all (the red book)
3/25/04
© University of Wisconsin, CS559 Spring 2004
OpenGL Commands (1)
• glMaterial{if}(face, parameter, value)
– Changes one of the coefficients for the front or back side of a face (or both
sides)
• glLight{if}(light, property, value)
– Changes one of the properties of a light (intensities, positions, directions, etc)
– There are 8 lights: GL_LIGHT0, GL_LIGHT1, …
• glLightModel{if}(property, value)
– Changes one of the global light model properties (global ambient light, for
instance)
• glEnable(GL_LIGHT0) enables GL_LIGHT0
– You must enable lights before they contribute to the image
– You can enable and disable lights at any time
3/25/04
© University of Wisconsin, CS559 Spring 2004
OpenGL Commands (2)
• glColorMaterial(face, mode)
– Causes a material property, such as diffuse color, to track the current
glColor()
– Speeds things up, and makes coding easier
• glEnable(GL_LIGHTING) turns on lighting
– You must enable lighting explicitly – it is off by default
• Don’t use specular intensity if you don’t have to
– It’s expensive - turn it off by giving 0,0,0 as specular color of the lights
• Don’t forget normals
– If you use scaling transformations, must enable GL_NORMALIZE to keep
normal vectors of unit length
• Many other things to control appearance
3/25/04
© University of Wisconsin, CS559 Spring 2004